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An impulsive nonlinear singular version of the Gronwall-Bihari inequality

Abstract

We find bounds for a Gronwall-Bihari type inequality for piecewise continuous functions. Unlike works in the prior literature, here we consider inequalities involving singular kernels in addition to functions with delays.

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References

  1. Bainov DD, Hristova SG: Impulsive integral inequalities with a deviation of the argument. Mathematische Nachrichten 1995, 171: 19–27.

    MathSciNet  Article  MATH  Google Scholar 

  2. Bainov DD, Simeonov PS: Systems with Impulse Effect: Theory and Applications. Ellis Horwood, Chichister; 1989.

    MATH  Google Scholar 

  3. Bainov DD, Simeonov PS: Integral Inequalities and Applications, Mathematics and Its Applications. Volume 57. Kluwer Academic, Dordrecht; 1992:xii+245.

    Book  MATH  Google Scholar 

  4. Butler G, Rogers T: A generalization of a lemma of Bihari and applications to pointwise estimates for integral equations. Journal of Mathematical Analysis and Applications 1971,33(1):77–81. 10.1016/0022-247X(71)90183-1

    MathSciNet  Article  MATH  Google Scholar 

  5. Hristova SG: Nonlinear delay integral inequalities for piecewise continuous functions and applications. Journal of Inequalities in Pure and Applied Mathematics 2004,5(4):1–14. article 88 article 88

    MathSciNet  MATH  Google Scholar 

  6. Kirane M, Tatar N-E: Global existence and stability of some semilinear problems. Archivum Mathematicum (Brno) 2000,36(1):33–44.

    MathSciNet  MATH  Google Scholar 

  7. Kirane M, Tatar N-E: Convergence rates for a reaction-diffusion system. Zeitschrift für Analysis und ihre Anwendungen. Journal for Analysis and Its Applications 2001,20(2):347–357.

    MathSciNet  Article  MATH  Google Scholar 

  8. Krylov NN, Bogolyubov NN: Introduction to Nonlinear Mechanics. Izd. Acad. Sci. Ukr. SSR, Kiev; 1937.

    Google Scholar 

  9. Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. Volume 6. World Scientific, New Jersey; 1989:xii+273.

    Book  Google Scholar 

  10. Mazouzi S, Tatar N-E: Global existence for some integro-differential equations with delay subject to non-local conditions. Zeitschrift für Analysis und ihre Anwendungen. Journal for Analysis and Its Applications 2002,21(1):249–256.

    MathSciNet  Article  MATH  Google Scholar 

  11. Medved M: A new approach to an analysis of Henry type integral inequalities and their Bihari type versions. Journal of Mathematical Analysis and Applications 1997,214(2):349–366. 10.1006/jmaa.1997.5532

    MathSciNet  Article  MATH  Google Scholar 

  12. Medved M: Singular integral inequalities and stability of semilinear parabolic equations. Archivum Mathematicum (Brno) 1998,34(1):183–190.

    MathSciNet  MATH  Google Scholar 

  13. Pachpatte BG: Inequalities for Differential and Integral Equations, Mathematics in Science and Engineering. Volume 197. Academic Press, California; 1998:x+611.

    MATH  Google Scholar 

  14. Samoilenko AM, Perestyuk NA: Stability of solutions of differential equations with impulse effect. Differential Equations 1977,13(11):1981–1992.

    Google Scholar 

  15. Samoilenko AM, Perestyuk NA: Impulsive Differential Equations, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises. Volume 14. World Scientific, New Jersey; 1995:x+462.

    Google Scholar 

  16. Tatar N-E: Exponential decay for a semilinear problem with memory. Arab Journal of Mathematical Sciences 2001,7(1):29–45.

    MathSciNet  MATH  Google Scholar 

  17. Zhang W, Agarwal RP, Akin-Bohner E: On well-posedness of impulsive problems for nonlinear parabolic equations. Nonlinear Studies 2002,9(2):145–153.

    MathSciNet  MATH  Google Scholar 

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Correspondence to Nasser-Eddine Tatar.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Tatar, NE. An impulsive nonlinear singular version of the Gronwall-Bihari inequality. J Inequal Appl 2006, 84561 (2006). https://doi.org/10.1155/JIA/2006/84561

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  • DOI: https://doi.org/10.1155/JIA/2006/84561

Keywords

  • Continuous Function
  • Type Inequality
  • Prior Literature
  • Piecewise Continuous Function
  • Singular Kernel